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Operations Research Models and Methods
Computation Section
Decision Analysis
 - Example
Computer Repair

A good customer has just given a microcomputer repairwoman a failed machine. She would like to fix it with the least expenditure of money. One of four parts is probably failed. They are A, B, C, and D, with the cost of replacement $100, $200, $30, and $80, respectively. We use the letters A through D to represent the events that the failure is caused by each of the four parts. Let E be the event that something other than the four parts is causing the computer failure.

Three tests, X, Y, and Z, are available to help locate the failed part. To perform the tests requires expenditures of $50, $70, and $80, respectively. If E is discovered or the repairwoman decides to abandon testing, the motherboard can be replaced at a cost of $500.

The goal of the analysis is to determine an optimum strategy maintenance that minimizes the expected cost of repair. The strategy will specify the tests to be performed and the actions that should be taken based on the outcomes of the tests.

Decision Node

In the following, we construct the decision tree that describes the details of the decision process. The decision tree is the model for decision analysis. We build the tree in steps, starting from the repairwoman's first decision.

When the repairwoman first gets the machine she can repair the machine by replacing the motherboard at a cost of $500 or use test X to get a better idea what's wrong. This represents the first decision in the process, whether to perform the test. In the figure at the left, we begin to construct the decision tree. The decision tree is a graphical description of a sequential process and constitutes the major part of model of the decision process. The first decision is indicated by the rectangular node labeled 1. The symbol D1 interior to the node identifies the decision. Two arcs leave the node indicating the two possible decisions available at this point. The first decision has two results: either replace the motherboard or perform the X test. The labels on the arcs 1 and 2 correspond to these two possibilities. The cost associated with the test (50) is shown adjacent to the arc entering node 3.

For this and the following decision nodes, we allow only the decisions specified in the text. For example, at D1 only two decisions are available, either replace the motherboard or do test X. Other conceivable actions such as replacing component A or the two components A and B are not allowed at D1. If they were, a much more complicated model would result.

Terminal Node

On the figure the decision to replace the motherboard leads to a terminal node, shown as a black circle, labeled node 2. At terminal nodes the process stops, and the cost associated with the terminal state can be evaluated. The number adjacent to the node (500) is the cost associated with reaching this node, that is, the cost of replacing the motherboard.
Chance Node

The decision to perform test X leads to node 3. This is a chance node because the result of the test is uncertain. This kind of node is shown as a white circle. Test X can come up with three indications :failure probably due to A or B (arc 3), failure probably due to C or D (arc 4), or an indication to replace the motherboard (arc 5).

The figure shows the three possible events associated with the experiment as arcs leaving the chance node. The numbers on the arcs leaving the node are the probabilities of the three test results. The events are mutually exclusive and constitute the entire range of possibilities, so their probabilities sum to 1. We describe how the probabilities are determined in a later section. The arcs terminate at nodes that either represent additional decisions or terminal events.

Note that this test does not accurately determine the cause of failure. For example if the test indicates that the failure is probably due to A or B, it could still be due to C, D or E. Only the probabilities are affected.

A Second Round of Decisions

If Test X shows the first outcome, the repairwoman can either use test Y or replace the motherboard. If the test shows the second outcome, she can use test Z or replace the motherboard. If the test shows the third outcome, she must replace the motherboard.

The figures show the detail associated with a second round of decisions. With the first outcome for test X, the repairwoman can either continue the fault isolation process with test Y or immediately replace the motherboard. This is shown as decision D2. The cost of test Y is $70. Note that only these two actions are allowed. She cannot at this point replace the individual components.

If test X results in the second outcome, she can continue the isolation with test Z or replace the motherboard. This is decision D3. The cost of test Z is $80. In both cases, the motherboard replacement costs $500.

Outcomes for Tests Y and Z

We assume for the example that test Y can accurately identify the cause of failure if it is due to component A or B. If either of these indications is observed, the faulty component is repaired. Every other cause (C, D, or E) is grouped into a third category. If the test does not indicate A or B, the motherboard is replaced.

Similarly, test Z accurately identifies the fault if it is due to C or D. If the test does not indicate C or D, the fault must be in A, B or E. Rather than continue testing, the motherboard is replaced. The test results together with the appropriate probabilities and costs are shown in the figures.


The Complete Tree

  The complete tree is shown in the figure below. The tree is the model of the decision process. It consists of numbered nodes and arcs. For a tree structure, the number of nodes is always one more than the number of arcs.

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tree roots

Operations Research Models and Methods
by Paul A. Jensen
Copyright 2004 - All rights reserved

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